Earth has strong magnetic field, which changes concentration of charged particles from Sun near the planet. There is rather safe torus near the Earth (LEO non-polar orbits), and also there is long tail, in which activity of Sun emitted particles is reduced (in this picture Sun is at left side and Earth is in center) http://en.wikipedia.org/wiki/File:Magnetosphere_Levels.svg

My question - what is geometry of magnetotail, if we check it 1.5 million kilometers away from Earth (at the line Sun-Earth-L2). What is estimated diameter of magnetopause or magnetotail here?

Will Earth provide extra protection from charged particles for the spacecraft hung exactly in L2 Lagrange point of the System Sun-Earth.

Will there substantial protection for the spacecraft orbiting around L2 at distance of 200,000 to 300,000 kilometers away from L2 (in the plane, perpendicular to the line Sun-Earth)

If the tail is depicted correctly, its diameter near L2 is ~1.5 times less then Moon's orbit diameter (0.77 mln km), or near 0.5 million kilometers.

The The Global Magnetosphere page from "Earth's Magnetosheath, Magnetopause and Magnetosphere" lecture by Iver Cairns, 1999 has interesting picture "Figure 14.7: Noon-midnight cross section of the magnetosphere and geomagnetic tail [Hughes, 1995]. Magnetic field lines are shown using solid lines while the dashed lines shows particles moving along the field subject to the ...", and I think that sizes are in R_e units (6400km).

200,000 - 300,000 km halo orbit around L2 is equal to 30-40 R_e. So I can conclude that Earth magnetosphere may cover the exact L2 point. But at the same time, it is unlikely to give much protection or constant protection to the satellite orbiting so far from L2 (actually, it is periapsis of halo orbit parameters of the Gaia).

$\begingroup$Check also Peter Delamere's book Magnetotails in the Solar System chapter 4 Earth's magnetotail page 62 figure 4.1 "The Earth's magnetotail projected into the noon-midnight meridian based on IMP-1 magnetic field data [ Ness, 1965]", "The tail is in fact a very dynamic structure with many internal processes and rapid changes [Sharma et al. 2008]"$\endgroup$
– osgxJan 11 '19 at 0:51